Palgrave Handbook of Econometrics: Applied Econometrics

714 Panel Methods to Test for Unit Roots and Cointegration

given by:

BCU=(INm−β(-β-′β)-−^1 β-′)B. (B.1)

The dimension of the space (spanned by)BCUis defined as the cross-unit cointe-
grating rank. The definition merely formalizes the notion that, considering here

the simplest example, cointegrating relationships of the form[β

′

1 ,β

′

2 ,0,...,0]

′that

involve variables from different cross-sections but lead to a stationary transformed

processβ

′

1 Y1,t+β

′

2 Y2,tmerely via combining already stationary processes from

different cross-section units (in the exampleβ

′

1 Y1,tandβ

′
2 Y2,t)should not be con-
sidered asgenuinecross-unit cointegrating relationships. The above definition of
the spaceBCUas the projection ofBon the ortho-complement ofβ-delivers all
cointegrating relationships that are not given by linear combinations of individ-

ual specific cointegrating relationships, that is,B=BCU⊕β-, with⊕denoting the
direct sum. We say that the panel exhibits long-run cross-sectional dependence if
the cross-unit cointegrating rank is larger than 0. In case the cross-sectional units
are correlated, but there is no cross-unit cointegration, we speak of (pure) short-run
cross-sectional dependence.
Note for completeness that the previous sentence implies that panels comprised
of independent random walks are short-run dependent – since independent ran-
dom walks are asymptotically correlated, this is the famous spurious correlation
(for example, Phillips, 1986). Therefore, if one were to be keen on excluding spu-
rious asymptotic correlations, the above definition of short-run dependence can
simply be reformulated and stated in terms of cross-sectional correlation once all
variables have been transformed to stationarity, that is, by first-order differenc-
ing all integrated variables in the panel. In other words, the definition might be
based on the innovations of the individual series to avoid the necessity to explicitly
discuss spurious correlations.
Given that we consider processes that have a representation as a solution to
autonomous stochastic difference equations, both forms of dependence, short-run
and long-run, originate in the dependence structure of the error processes driving
the individual series in the respective difference equations. Consider, for example,
the DGP considered for testing for cointegration by single equation methods in
(13.18)–(13.22), abstracting here from deterministic components. In this system
the stochastic behavior is governed by the three random processesηt,εi,tandvi,t.
Theraison d’êtreof the common factors is to induce cross-sectional dependencies
via the common factors, but short- and long-run dependence can also arise via the
other two components as soon as the assumption that they are cross-sectionally
independent is relaxed. To illustrate the issue consider (13.21), that is,( 1 −φiL)ei,t=
Hi(L)εi,t, with theεi,tbeing white-noise processes. Noting that the fact that the
εi,tare individually white-noise processes does not imply that the stacked vector

εt=

(

ε1,t,...,εN,t

)′

is a vector white-noise process, it becomes immediately clear